# 07: Work and Kinetic Energy

### Check Your Understanding

**7.1. **No, only its magnitude can be constant; its direction must change, to be always opposite the relative displacement along the surface.

**7.2. **No, it’s only approximately constant near Earth’s surface.

**7.3. **W = 35 J

**7.4.** a. The spring force is the opposite direction to a compression (as it is for an extension), so the work it does is negative. b. The work done depends on the square of the displacement, which is the same for x = ± 6 cm, so the magnitude is 0.54 J.

**7.5. **a. The car; b. the truck

**7.6.** Against

**7.7.** 3 m/s

**7.8. **980 W

### Conceptual Questions

**1. **When you push on the wall, this “feels” like work; however, there is no displacement so there is no physical work. Energy is consumed, but no energy is transferred.

**3.** If you continue to push on a wall without breaking through the wall, you continue to exert a force with no displacement, so no work is done.

**5. **The total displacement of the ball is zero, so no work is done.

**7.** Both require the same gravitational work, but the stairs allow Tarzan to take this work over a longer time interval and hence gradually exert his energy, rather than dramatically by climbing a vine.

**9.** The first particle has a kinetic energy of 4(\(\frac{1}{2}\)mv^{2}) whereas the second particle has a kinetic energy of 2(\(\frac{1}{2}\)mv^{2}), so the first particle has twice the kinetic energy of the second particle.

**11.** The mower would gain energy if −90° < \(\theta\) < 90°. It would lose energy if 90° < \(\theta\) < 270°. The mower may also lose energy due to friction with the grass while pushing; however, we are not concerned with that energy loss for this problem.

**13. **The second marble has twice the kinetic energy of the first because kinetic energy is directly proportional to mass, like the work done by gravity.

**15.** Unless the environment is nearly frictionless, you are doing some positive work on the environment to cancel out the frictional work against you, resulting in zero total work producing a constant velocity.

**17.** Appliances are rated in terms of the energy consumed in a relatively small time interval. It does not matter how long the appliance is on, only the rate of change of energy per unit time.

**19. **The spark occurs over a relatively short time span, thereby delivering a very low amount of energy to your body.

**21.** If the force is antiparallel or points in an opposite direction to the velocity, the power expended can be negative.

### Problems

**23. **3.00 J

**25.** a. 593 kJ

b. –589 kJ

c. 0

**27.** 3.14 kJ

**29.** a. –700 J

b. 0; c. 700 J

d. 38.6 N

e. 0

**31.** 100 J

**33. **a. 2.45 J

b. – 2.45 J

c. 0

**35. **a. 2.22 kJ

b. −2.22 kJ

c. 0

**37.** 18.6 kJ

**39.** a. 2.32 kN

b. 22.0 kJ

**41.** 835 N

**43. **257 J

**45. **a. 1.47 m/s

b. Answers may vary

**47. **a. 772 kJ

b. 4.0 kJ

c. 1.8 x 10^{−16} J

**49.** a. 2.6 kJ

b. 640 J

**51. **2.72 kN

**53. **102 N

**55. **2.8 m/s

**57. **W(bullet) = 20 x W(crate)

**59.** 12.8 kN

**61. **0.25

**63. **a. 24 m/s, −4.8 m/s^{2}

b. 29.4 m

**65. **310 m/s

**67. **a. 40

b. 8 million

**69. **$149

**71. **a. 208 W

b. 141 s

**73.** a. 3.20 s

b. 4.04 s

**75. **a. 224 s

b. 24.8 MW

c. 49.7 kN

**77. **a. 1.57 kW

b. 6.28 kW

**79.** 6.83 \(\mu\)W

**81.** a. 8.51 J

b. 8.51 W

**83. **1.7 kW

### Additional Problems

**85. **15 N • m

**87.** 39 N • m

**89. **a. 208 N • m

b. 240 N • m

**91.** a. −0.9 N • m

b. −0.83 N • m

**93.** a. 10. J

b. 10. J

c. 380 N/m

**95.** 160 J/s

**97.** a. 10 N

b. 20 W

### Challenge Problems

**99.** If crate goes up: a. 3.46 kJ

b. −1.89 kJ

c. −1.57 kJ

d. 0

**100. I**f crate goes down: a. −0.39 kJ

b. −1.18 kJ

c. 1.57 kJ

d. 0

**101. **8.0 J

**103. **35.7 J

**105.** 24.3 J

**107. **a. 40 hp

b. 39.8 MJ, independent of speed

c. 80 hp, 79.6 MJ at 30 m/s

d. If air resistance is proportional to speed, the car gets about 22 mpg at 34 mph and half that at twice the speed, closer to actual driving experience.

### Contributors

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).